Question

A random sample of 390 married couples found that 298 had two or
more personality preferences in common. In another random sample of
582 married couples, it was found that only 28 had no preferences
in common. Let *p*_{1} be the population proportion
of all married couples who have two or more personality preferences
in common. Let *p*_{2} be the population proportion
of all married couples who have no personality preferences in
common.

(a) Find a 95% confidence interval for *p*_{1} –
*p*_{2}. (Round your answers to three decimal
places.)

lower limit | |

upper limit |

(b) Explain the meaning of the confidence interval in part (a) in
the context of this problem. Does the confidence interval contain
all positive, all negative, or both positive and negative numbers?
What does this tell you (at the 95% confidence level) about the
proportion of married couples with two or more personality
preferences in common compared with the proportion of married
couples sharing no personality preferences in common?

Because the interval contains only positive numbers, we can say that a higher proportion of married couples have two or more personality preferences in common.We can not make any conclusions using this confidence interval. Because the interval contains both positive and negative numbers, we can not say that a higher proportion of married couples have two or more personality preferences in common.Because the interval contains only negative numbers, we can say that a higher proportion of married couples have no personality preferences in common.

Answer #1

A) = 298/390 = 0.764

= 28/582 = 0.048

The pooled sample proportion (P) = ( * n1 + * n2)/(n1 + n2)

= (0.764 * 390 + 0.048 * 582)/(390 + 582)

= 0.335

SE = Sqrt(P(1 - P)(1/n1 + 1/n2))

= sqrt(0.335 * (1 - 0.335) * (1/390 + 1/582))

= 0.031

At 95% confidence interval the critical value is z^{*} =
1.96

The 95% confidence interval is

()
+/- z^{*} * SE

= (0.764 - 0.048) +/- 1.96 * 0.031

= 0.716 +/- 0.061

= 0.655, 0.777

Because the interval cointains only positive numbers , we can say that a higher proportion of married couples have two or more personality preference in common.

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